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			== Random_variable ==
			<gallery>
			File:Random_Variable_as_a_Function-en.svg\|Random Variable as a Function
			File:Dice_Distribution_(bar).svg\|Dice Distribution (bar)
			</gallery>

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'''Random variable'''

A '''random variable''' is a variable whose possible values are outcomes of a random phenomenon. More specifically, a random variable is defined as a function that maps the outcomes of an unpredictable process to numerical quantities, typically real numbers, which can be either discrete (i.e., taking any of a specified list of exact values) or continuous (i.e., taking any numerical value in an interval or collection of intervals).

== Definition ==

Formally, let ''S'' be a [[Sample space|sample space]] — a set of outcomes of a random process. Then a '''random variable''' ''X'' is a function from ''S'' to the real numbers. The technical conditions on the function ''X'' are designed to capture the idea that the value of ''X'' is determined by the outcome of the random process.

== Types of random variables ==

Random variables can be classified into two broad types:

* '''[[Discrete random variable]]s''': These are variables that can take on a countable number of values. Examples include the number of heads in a series of coin flips, or the number of students present in a classroom.

* '''[[Continuous random variable]]s''': These are variables that can take on any value in a given range or interval. Examples include the height of a person, or the time it takes for a computer to process a task.

== Probability distribution ==

The '''[[Probability distribution]]''' of a random variable is a function that describes the likelihood of each possible outcome. It is defined for every possible outcome in the sample space and satisfies the conditions of a probability measure.

== Expectation and variance ==

The '''[[Expected value]]''' or '''expectation''' of a random variable is a weighted average of all possible values that the random variable can take on, with the weights being the probabilities of these outcomes. The '''[[Variance]]''' of a random variable is a measure of how much values of the random variable vary around the expected value.

== See also ==

* [[Probability theory]]
* [[Statistics]]
* [[Stochastic process]]
* [[Probability density function]]
* [[Cumulative distribution function]]

== References ==

<references />

[[Category:Probability theory]]
[[Category:Statistics]]
[[Category:Stochastic processes]]

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Random variable - Revision history

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